## Abstract

Let $\U0001d524$ be a simple Lie algebra. Let $\mathrm{Aut}(\U0001d524)$ be the group of all automorphisms on $\U0001d524$, and let $\mathrm{Int}(\U0001d524)$ be its identity component. The outer automorphism group of $\U0001d524$ is defined as $\mathrm{Aut}(\U0001d524)/\mathrm{Int}(\U0001d524)$. If $\U0001d524$ is complex and has Dynkin diagram $\mathrm{D}$, then $\mathrm{Aut}(\U0001d524)/\mathrm{Int}(\U0001d524)$ is isomorphic to $\mathrm{Aut}(\mathrm{D})$. We provide an analogous result for the real case. For $\U0001d524$ real, we let $\U0001d524$ be represented by a painted diagram $\mathrm{P}$. Depending on whether the Cartan involution of $\U0001d524$ belongs to $\mathrm{Int}(\U0001d524)$, we show that $\mathrm{Aut}(\U0001d524)/\mathrm{Int}(\U0001d524)$ is isomorphic to $\mathrm{Aut}(\mathrm{P})$ or $\mathrm{Aut}(\mathrm{P})\times {\mathbb{Z}}_{2}$. This result extends to the outer automorphism groups of all real semisimple Lie algebras.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.